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I am running auto.arima/ETS models from the forecast package in R on monthly seasonal time series. I see the following fitted picture. Blue - actual data, red - the fit.
I see that the fit is not capturing the series well - specifically it is not capturing well the peaks and foots. But the model is recognizing the general curvature of the series.

Question: any suggestions how to tune the model to get a better fit?

enter image description here

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A model with a better fit =/= a model with better forecasts in the time series world. Most 'auto' tuning frameworks optimize based on an approximation of the forecast accuracy rather than in-sample fit. If you want a good fit from an arima you could grid search the parameters to minimize in-sample rmse, but you will overfit.

If there is a reason for the peaks and you want to capture it then you could pass variables to account for them, otherwise the model smoothing them out is the most reasonable thing to do.

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